In front of a capacity crowd at Scotstoun, the Warriors continued their unbeaten start to the season with a monumental victory over reigning RaboDirect PRO12 champions, Leinster.
Tries from Fusaro and Yanuyanutawa allowed Glasgow to win their opening three games for the first time since 2002.
In a turgid first half, the Warriors ground down a determined visiting side to lead at half time before exerting pressure and grinding out a famous win.
Townsend made a number of changes from the side that defeated Ulster at the death last week.
Chris Cusiter made his first start for the club since September 2012 after spending almost a year on the sidelinesa.
The Warriors coach was forced into a late change with the try scorer against Cardiff Blues, Tommy Seymour, replaced DTH van der Merwe who fell ill before kick off.
Leinster also had a surprise inclusion in their starting 15 with Aussie giant, Lote Tuqiri starting on the wing. The Wallaby didn’t last long though, being replaced by Rob Kearney on the half hour mark.
In what was a scrappy opening, both teams were unable to make any decisive line breaks with Leinster proving to be a streetwise side and defensively strong.
After 23 minutes, Leinster were awarded their first kickable penalty close to the half way line and Leinster fly half, Jimmy Gopparth made no mistake in putting the away side 3-0 up.
Shortly after, Glasgow were awarded a penalty of their own on the right flank but Stuart Hogg skewed the difficult kick wide of the posts.
The British and Irish Lions back had a frustrating first half in front of the sticks and missed a penalty after 32 minutes from the halfway line.
Leinster suffered a set back before half time with their captain, Shane Jennings having to retire injured.
Despite a turgid first period, the Warriors battled hard and notched a try with two minutes remaining. Having pressured the Leinster line, the home side were rewarded by Chris Fusaro who crossed the whitewash.
Stuart Hogg then added the extras shortly before the half time whistle.
Half time: Glasgow Warriors 7-3 Leinster
Early in the second period Mark Bennett stepped up to take a penalty after 50 minutes for Glasgow. The centre gave the ball an almighty thump but saw it agonisingly hit the under side of the crossbar.
After 59 minutes, Leinster notched a penalty from distance with McGuigan and Strauss replacing Hogg and Fusaro shortly after the ball went over the sticks.
With just one point in it, Glasgow piled forward in front of a raucous support and were rewarded with a penalty wide on the right.
Ruaridh Jackson stepped up and smacked it wide of the target.
The game was on a knife edge as the final whistle neared with Glasgow pouring forward as Warrior Nation rose to their feet to cheer on the team.
As they mounted pressure on the Leinster try line, substitute, Yanuyanutawa rampaged his way over the whitewash to give the Warriors a six point cushion going into the dying moments.
The home side held on to secure their first home victory over Leinster in three years.